If the center of a regular hexagon is at the origin and one of the vertices on the Argand diagram is $1 + 2i$,then its perimeter is

Let $z_{1}$ be a fixed point on the circle of radius $1$ centered at the origin in the Argand plane and $z_{1} \neq \pm 1$. Consider an equilateral triangle inscribed in the circle with $z_{1}, z_{2}, z_{3}$ as the vertices. Then, $z_{1} z_{2} z_{3}$ is equal to

The region of the complex plane for which $\left| \frac{z - a}{z + \overline{a}} \right| = 1$ where $\text{Re}(a) \neq 0$ is

The number of points $z$ on the Argand plane which satisfy the conditions $\operatorname{Re}\left(\frac{z-2}{z-4i}\right)=0$ and $\operatorname{Im}\left(\frac{z-2}{z-4i}\right)=1$ simultaneously is

For $z \in \mathbb{C}$,if the minimum value of $(|z-3 \sqrt{2}| + |z-p \sqrt{2} i|)$ is $5 \sqrt{2}$,then a value of $p$ is $.......$

The minimum value of $|z-1|+|z-5|$ is